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Finding one root $x_1$ of the quintic equation $x^5 + x = -a$ by using the Bring radical is described on Wikipedia.

The root is $x_1 = -a +a^5 -5a^9+35a^{13}+ \ldots$ , and it is found by reversion of the Taylor series for $f(x) = x^5 + x$.

How do we find the other roots of this quintic in series representation?

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The answer is contained in the paper by Eagle (1939) http://www.jstor.org/stable/2303036?seq=1#page_scan_tab_contents

The Bring-Jerrard normal form is a particular example of a trinomial, and Eagle gives the general solution.

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